Recently, Marcus, Spielman, and Srivastava proved the existence of infinite families of bipartite Ramanujan graphs of every degree at least 3 by using the method of interlacing families of polynomials. In this talk, we apply their method to prove that for any connected graph G, there exists an orientation of G such that the spectral radius of the corresponding Hermitian adjacency matrix is at most that of the universal cover of G.

A fractional matching of a graph G is a function f giving each edge a number between 0 and 1 so that \(\sum_{e \in \Gamma(v)} f(e) \le 1\) for each \(v \in V(G)\), where \(\Gamma(v)\) is the set of edges incident to v. The fractional matching number of G, written \(\alpha’_*(G)\), is the maximum of \(\sum_{e \in E(G)} f(e)\) over all fractional matchings f. Let G be an n-vertex graph with minimum degree d, and let \(\lambda_1(G)\) be the largest eigenvalue of G. In this talk, we prove that if k is a positive integer and \(\lambda_1(G) < d\sqrt{1+\frac{2k}{n-k}}[/latex], then [latex]\alpha'_*(G) > \frac{n-k}{2}\).

A Halin graph is constructed from a plane embedding of a tree whose non-leaf vertices have degree at least 3 by adding a cycle through its leaves in the natural order determined by the embedding. In this talk, we prove that every 3-connected \(\{K_{1,3},P_5\}\)-free graph has a spanning Halin subgraph. This result is best possible in the sense that the statement fails if \(K_{1,3}\) is replaced by \(K_{1,4}\) or \(P_5\) is replaced by \(P_6\). This is a joint work with Guantao Chen, Jie Han, Songling Shan, and Shoichi Tsuchiya.

Path Cover Number in 4-regular Graphs and Hamiltonicity in Connected Regular Graphs

Suil O (오수일)
Department of Mathematics, The College of William and Mary, Williamsburg, Virginia, USA

2012/5/16 Wed 4PM

A path cover of a graph is a set of disjoint paths such that every vertex in the graph appears in one of the paths. We prove an upper bound for the minimum size of a path cover in a connected 4-regular graph with n vertices, confirming a conjecture by Graffiti.pc. We also determine the minimum number of vertices in a connected k-regular graph that is not Hamiltonian, and we solve the analogous problem for Hamiltonian paths.
This is a partly joint work with Gexin Yu and Rui Xu.